Major League Baseball Proposes A New Extra Innings Rule–Let’s Look At The Math – Forbes

“Hey guys. How much longer you think this game is gonna last?” (Photo by Dilip Vishwanat/Getty Images)

Baseball games are long sometimes. For those of us who really love baseball this is part of the charm, but for the average mid-week ballgame attendee who has to go to work in the morning it might be nice if things could move along a bit quicker. And heaven forbid the game go into extra innings.

Baseball is the unique sport that doesn’t put a time limit on its games. Other sports have overtime periods, but they are time-limited such as in basketball and hockey, or they are effectively decided by a coin toss (I’m looking at you NFL). Not so with baseball. In theory a game could go on forever, although the longest game in MLB history was a mere 26 innings in 1920 (the Brooklyn Dodgers were visiting the Boston Red Sox that day) and by far the most likely outcome for an extra innings game is that it ends after 10. As Philip Bump breaks it down in the Washington Post on February 10, 2017, that means that a typical extra innings game is only 20-40 minutes longer than a regular 9-inning game. (Bump’s article has a lot of good data and plots that you should check out.)

But Major League Baseball, ever worried about these sorts of things, has decided to try an experiment. In a couple of rookie leagues this year extra innings games will have a new twist: each team starts each extra inning with a runner on second. My knee-jerk reaction to this is one of horror. In fact, in the interest of full disclosure I’ll go on record as saying I still despise the designated hitter. And I long for the days of my youth when teams played scheduled double-headers and there were no giant Jumbotrons blaring inane games and ads between innings. But I digress.

Will this change actually make a difference? Bump’s time calculation suggests not much, but I want to consider the likelihood that this will actually make the game last fewer innings. The crucial question is this: given that a team has a runner on second with no outs, what is the probability that the team will score at least one run? Luckily, baseball data is easy to find and analyze. Based on calculations from every game between 1984 and 1994 (data available here) the expected number of runs and the probability of scoring is shown in the following table.

Explanation: the “Runners” column indicates the location of base runners; so the first row is bases empty and the second is a single runner on second. The “Expected” column is the expected number of runs the team will score in that particular scenario. “Prob > 0” is the probability that a team will score at least one run in the inning. A caveat: these numbers were computed using only 9-inning games (no extra innings), but they should be fairly representative of what one can expect in practice. Of course, strange things happen in extra innings, especially if the game goes on long enough and a random position player ends up pitching.

The thing to notice is that the odds are much better that a team will score if it starts with a runner on second and no outs. Duh. That’s MLB’s whole rationale for this in the first place. But the problem is that both teams are more likely to score, effectively canceling out the advantage, right? Well, let’s do some calculations. How can a game end after 10 innings? There are three possibilities: (1) the visiting team scores at least one run and the home team scores fewer in the bottom of the frame; (2) the visiting team does not score and the home team scores at least one run; (3) the home team scores more runs in the inning than the visiting team. Under the current system, the first possibility happens with probability roughly 0.275 × 0.725 = 0.199375 and the same is true for possibility (2). It’s trickier to estimate the likelihood of the third scenario, but luckily the probability of scoring a particular number of runs in an inning has been calculated (p. 130); the results are in this table.

So the probability of the third event is the sum of the probabilities that the visiting team scores i runs while the home team scores i + 1 runs for i > 0. This comes to only 0.002626. So the total probability that a game ends after 10 innings is approximately 40.14%.  And historical data back this up: Since 1920 44% of extra innings games ended after 10 innings. And another 25% of extra innings games end after 11.

So what’s the probability that a game ends after 10 innings under the new scheme? Now the probabilities of the first two scenarios are both 0.633 × 0.367 = 0.232311. As above, the probability of the third event is negligible and so we can expect a 46.46% chance a game will end after 10 innings. Let’s round that up to 50% just to be safe.

To sum up: this rather drastic rule change, one that flies in the face of 150 years of tradition, will give MLB about a 6% increase in the likelihood of ending a game after 10 innings. Similar analysis tells us that we can expect about 75% of games to end in no more than 11 innings while the current rate is about 69%. Doesn’t seem worth it, does it?